A refined looped-functional for discrete-time Markov jump systems under aperiodic sampling

The looped-functional and its corresponding stability analysis approach have proven highly effective for continuous-time sampled-data systems. However, for discrete-time linear systems (DLS), the looped-function approach is still lacking. A discrete looped-functional approach is proposed to investigate the asymptotic stability of DLS with Markov parameters and aperiodic sampling. Firstly, the discrete-time sampled-data system is represented as a time-delay system by using the input delay approach. Furthermore, a relaxed equivalent asymptotic stability condition is proposed, which only requires that the system state tends to zero at ordered and non-adjacent discrete instants $\left\{t_k\right\}$, which effectively simplify the complexity of the stability analysis. Moreover, for discrete-time systems under aperiodic sampling, based on the above equivalent condition, a novel discrete looped-functional is constructed. As an extension of the continuous looped-functional, the discrete looped-functional has the characteristic that its value is zero at $\left\{t_k\right\}$ and it does not impose strict constraint on the positive definiteness of the functional. By utilizing the less conservative looped-functional and the summation inequality, two asymptotic stability conditions are established. The validity of these results is verified by using a classical example, demonstrating a larger upper bound for the sampling interval compared to existing results.